The win ahead of Mexican team mate Sergio Perez completed a hat-trick for Verstappen.
Seven-time world champion Lewis Hamilton and the FIA were in a standoff at this month's Miami Grand Prix over piercings the Mercedes driver has raced with for years and said he could not remove.
The victory was the Dutch driver's third of the season, and second in a row, and he made it happen with a crucial overtake around the outside on Sainz at the start to seize second place before blasting past Leclerc on the ninth of the 57 laps.
Formula One and Liberty Media are self-promoting the race with their Live Nation entertainment company and local stakeholders.
Verstappen has won at Imola for the second year in a row.
Leclerc's car was dominant on the upgraded Albert Park circuit as the Monagesque sealed his fourth career win more than 20 seconds ahead of Red Bull's runner-up Sergio Perez.
Vettel is something of an Australian Grand Prix specialist, having won the race three times.
The sport's only Black driver, an outspoken champion for equal opportunities and diversity, Hamilton has also experienced racism and online abuse.
The Dutchman crossed the line just half-a-second ahead of Leclerc, who keeps the championship lead.
The Monegasque, who started from pole, crossed the line 5.5 seconds ahead of Spaniard Sainz.